Group #19133283 — subgroups

Subgroups — #19133283

Order 5504.

216 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583456	C₂	1	1	yes	maximal, characteristic
2	#14582923	C₂	1	1	yes	maximal, characteristic
3	#14582647	C₂	1	2	yes	maximal, complemented
4	#14582646	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#10621877	—	2	1	no	—
7	#13063325	C₂²	1	1	yes	characteristic
8	#13063325	C₂²	1	1	yes	characteristic
9	#13063317	—	2	1	no	—
10	#13063304	—	2	1	no	—
11	#13063304	C₄	1	2	yes	—
12	C₄×Dic₈₆	—	2	1	no	—
13	C₄×Dic₈₆	C₄	1	2	yes	—
14	#13063284	—	2	1	no	—
15	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
16	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
17	C₂²×C₈×C₄₃	—	2	1	no	—
18	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₄₃	—	2	1	no	—
20	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
21	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₂×Dic₈₆	—	2	1	no	—
25	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₄₃	—	2	2	no	—
27	C₂×C₈×C₄₃	—	4	2	no	—
28	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₄₃	D₄	1	1	yes	characteristic
30	C₄²×C₄₃	D₄	1	1	yes	characteristic
31	C₄²×C₄₃	C₂³	1	1	yes	characteristic
32	C₂²⋊Dic₄₃	—	2	2	no	—
33	C₄⋊Dic₄₃	—	2	1	no	—
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	—	2	1	no	—
36	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
37	Dic₄₃⋊C₄	—	2	2	no	—
38	Dic₄₃⋊C₄	—	2	2	no	—
39	C₄×Dic₄₃	—	2	2	no	—
40	#10621877	—	4	2	no	—
41	C₂×C₄₃⋊C₈	—	2	2	no	—
42	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₃	—	2	1	no	—
44	C₂×C₄×C₄₃	—	2	1	no	—
45	C₂×C₄×C₄₃	—	4	1	no	—
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
51	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₄₃	—	2	2	no	—
55	C₂×Dic₄₃	—	2	2	no	—
56	Dic₈₆	—	4	1	no	—
57	Dic₈₆	—	4	1	no	—
58	Dic₈₆	—	4	2	no	—
59	C₈×C₄₃	—	4	2	no	—
60	C₄₃⋊C₈	—	4	2	no	—
61	C₂²×C₄₃	—	2	1	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	#103	1	2	yes	—
64	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₄₃	—	2	1	no	—
66	C₄×C₄₃	—	2	1	no	—
67	C₄×C₄₃	—	2	1	no	—
68	C₄×C₄₃	#103	1	2	yes	—
69	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₄₃	#137	1	1	yes	characteristic
73	Dic₄₃	—	4	2	no	—
74	Dic₄₃	—	4	2	no	—
75	#1550	—	43	1	no	maximal, Sylow
76	C₂×C₄₃	—	2	2	no	—
77	C₂×C₄₃	#413	1	1	yes	characteristic
78	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
79	C₂×C₄₃	#417	1	1	yes	characteristic
80	#533	—	43	1	no	—
81	#423	—	43	1	no	—
82	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic
83	#333	—	43	2	no	—
84	#332	—	43	2	no	—
85	C₄₃	#1550	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	86	1	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₄⋊Q₈	—	43	1	no	—
89	C₄⋊Q₈	—	43	1	no	—
90	#125	—	86	1	no	—
91	C₂²⋊Q₈	—	43	2	no	—
92	C₂²⋊Q₈	—	86	1	no	—
93	C₄×Q₈	—	43	2	no	—
94	C₄×Q₈	—	86	1	no	—
95	#117	—	86	1	no	—
96	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	43	2	no	—
99	C₄⋊C₈	—	86	1	no	—
100	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
101	C₂×Q₈	—	43	2	no	—
102	C₂×Q₈	—	86	1	no	—
103	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
104	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
106	M₁₆	—	172	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	86	2	no	—
110	C₄⋊C₄	—	43	2	no	—
111	C₄⋊C₄	—	86	2	no	—
112	C₄⋊C₄	—	86	2	no	—
113	C₄⋊C₄	—	86	1	no	—
114	C₄⋊C₄	—	86	1	no	—
115	C₄⋊C₄	—	86	1	no	—
116	C₂²⋊C₄	—	86	2	no	—
117	C₄²	—	86	2	no	—
118	C₄²	C₄×D₄₃	1	1	yes	characteristic
119	C₄²	D₁₇₂	1	1	yes	characteristic
120	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
121	C₄²	C₂²×D₄₃	1	1	yes	characteristic
122	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
123	Q₈	—	172	2	no	—
124	Q₈	—	172	1	no	—
125	Q₈	—	172	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	86	2	no	—
130	C₂×C₄	—	86	2	no	—
131	C₂×C₄	#10621873	1	2	yes	—
132	C₂×C₄	Dic₁₇₂	1	2	yes	—
133	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
137	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₈	—	172	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#13063250	1	2	yes	—
144	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	172	2	no	—
149	C₄	—	172	2	no	—
150	C₄	#13063250	1	2	yes	—
151	C₄	C₂×#10621873	1	1	yes	characteristic
152	C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
153	C₄	#13063351	1	1	yes	characteristic
154	C₄	#13063356	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#14582881	1	1	yes	characteristic
157	C₂	C₂×#13063250	1	1	yes	characteristic
158	C₂	#14582898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583456

C₂

yes

maximal, characteristic

#14582923

C₂

yes

maximal, characteristic

#14582647

C₂

yes

maximal, complemented

#14582646

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic

C₂×#10621877

—

#13063325

C₂²

yes

characteristic

#13063325

C₂²

yes

—

—

C₄

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₄

yes

—

#13063284

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

Q₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#103

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#103

yes

—

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

C₂×Q₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

#1550

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

C₂×C₄⋊C₈

D₄₃

yes

characteristic

#333

—

#332

—

C₄₃

#1550

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₄₃

yes

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂²×C₄

C₄×D₄₃

yes

characteristic

104

C₂²×C₄

D₁₇₂

yes

characteristic

105

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

106

M₁₆

—

172

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₄×D₄₃

yes

characteristic

119

C₄²

D₁₇₂

yes

characteristic

120

C₄²

C₂²⋊D₄₃

yes

characteristic

121

C₄²

C₂²×D₄₃

yes

characteristic

122

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

123

Q₈

—

172

—

124

Q₈

—

172

—

125

Q₈

—

172

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#10621873

yes

—

132

C₂×C₄

Dic₁₇₂

yes

—

133

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

135

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

136

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

137

C₂×C₄

C₂×D₁₇₂

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

139

C₈

—

140

C₈

—

172

—

141

C₂²

—

142

C₂²

—

143

C₂²

#13063250

yes

—

144

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

172

—

149

C₄

—

172

—

150

C₄

#13063250

yes

—

151

C₄

C₂×#10621873

yes

characteristic

152

C₄

C₂×Dic₁₇₂

yes

characteristic

153

C₄

#13063351

yes

characteristic

154

C₄

#13063356

yes

characteristic

155

C₂

—

156

C₂

#14582881

yes

characteristic

157

C₂

C₂×#13063250

yes

characteristic

158

C₂

#14582898

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#1550
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₃)⋊C₄

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₂×C₄⋊C₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄⋊C₈

D₄₃

hypocenter

C₄₃

#1550

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #14582647	C₂	semidirect
4. #14582646	C₂	semidirect
85. C₄₃	#1550	semidirect

H ≅ G/N

type

3. #14582647

C₂

semidirect

4. #14582646

C₂

semidirect

85. C₄₃

#1550

semidirect

Subgroups — #19133283

Conjugacy classes of subgroups

Complements